Testing self-energy embedding theory in combination with GW

Testing self-energy embedding theory in combination with GW Tran Nguyen Lan,1, 2, ∗ Avijit Shee,1 Jia Li,2 Emanuel Gull,2 and Dominika Zgid1

arXiv:1706.05774v1 [cond-mat.str-el] 19 Jun 2017

1

Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, USA 2 Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA (Dated: June 20, 2017)

We present a theoretical framework and implementation details for self-energy embedding theory (SEET) with the GW approximation for the treatment of weakly correlated degrees of freedom and configuration interactions solver for handing the strongly correlated degrees. On a series of molecular examples, for which the exact results are known within a given basis, we demonstrate that SEET(CI/GW) is a systematically improvable and well controlled method capable of giving accurate results and well behaved causal self-energies and Green’s functions. We compare the theoretical framework of SEET(CI/GW) to that of GW+DMFT and comment on differences between these to approaches that aim to treat both strongly and weakly correlated simultaneously.

I.

INTRODUCTION

The quantitative simulation of realistic correlated solids and molecules requires the accurate description of a large number of degrees of freedom. Only some of these are strongly correlated. A successful approach has therefore been the combination of weak correlation methods, such as the GW method or the density functional theory in the local density approximation (LDA), with non-perturbative methods for low-energy effective models, such as the dynamical mean field approximation. The polynomial scaling of the weak correlation method then allows one to treat large systems, while the dynamical mean field approximation replaces the intractable lattice problem with an impurity problem coupled to a selfconsistently adjusted bath, which is numerically solvable. The combination of band structure methods with dynamical mean field theory (DMFT)1,2 has enjoyed great success, in particular when applied to materials with d and f shells. However, ambiguities at the interface of the weak- and strong-correlation methods, related in particular to the choice of “double counting” and the impurity interaction parameters in LDA+DMFT3 , and to the proper numerical treatment of general four-fermion screened interaction terms in GW+DMFT4–6 , have hampered progress. Moreover, the assumption that strong correlations are spatially localized, which is the premise of the dynamical mean field approximation, may not be valid in real compounds. Recently we introduced an approximation scheme, the self-energy embedding theory (SEET)7–12 that does not suffer from these limitations. First, due to the diagrammatic nature of the method, no double counting problem arises. Second, the absence of frequency-dependent interactions in the strongly correlated part means that standard multi-orbital impurity solvers can be employed and no ambiguity in how the Coulomb interactions are handled exist. Finally, the method does not rely on an a priori determination of the correlated subspace and in particular does not assume that strong correlations are local, but rather introduces a small control parameter that can be used to adaptively choose orbitals and sys-

tematically converge to the exact solution. In a series of previous papers7–12 , we studied the behavior of SEET where the weak-correlation method employed was the second order perturbation theory (GF2). In the present paper, we show how SEET performs when GF2 is replaced by the GW approximation. QM/QM embedding methods for realistic systems are not well studied, and prior work on non-perturbative strong correlation methods for small systems based on diagrammatic theory have shown problems ranging from convergence to the “wrong” fixed point13 to causality violations14 . We show here that no such problems are observed in our implementation of SEET, and that in fact the precision achieved with SEET along the entire range from weak to strong correlations is comparable to state-of-the-art quantum chemistry wave function methods, to which we carefully compare. As a testbed, we use small molecular examples, for which exact or nearly exact reference results within a given basis set are available. The remainder of this paper proceeds as follows. In section II, we briefly discuss the theoretical setup and the SEET functional. We then follow with benchmarking and in depth explanation of the numerical results in Sec. III. Finally, we present a theoretical discussion supporting our results and conclude in Sec. V. II.

METHODS

In this paper, we study small molecular systems. The specification of the interatomic distance and of a finite set of N gaussian orbitals fully determines the Hamiltonian in second quantized form, H=

N X ij

tij a†i aj +

N X

vijkl a†i a†j al ak ,

(1)

ijkl

where the operators a†i (ai ) create (destroy) an electron in orbital i, tij denotes the single-particle contribution, and vijkl the Coulomb matrix element. We express physical properties such as energies and single-particle response functions in a statistical mechanics approach based on an approximation to the grand

2 partition function Z = Tre−β(H−µn) . Here, β denotes the inverse temperature, µ the chemical potential, and n the density operator. Our temperatures are chosen low enough that the system has converged to the ground state, and the chemical potential µ is adjusted to yield the correct particle density. Within this framework, a functional Φ[G] of the Green’s function G, which contains all linked closed skeleton diagrams,15 is used to express the grand potential as

perturbation theory and does not recover the correct dissociated limit. Precise calculations of the second-order results require several numerical optimizations. Our implementation makes use of adaptive grids for both imaginary time20 and imaginary frequency21 Green’s functions.

Ω = Φ − Tr(log G−1 ) − Tr(ΣG),

The self-consistent GW method28,29 is a diagrammatic approximation formulated in terms of renormalized propagators G and renormalized (“screened”) interactions W . Similar to GF2, it can be written as an approximation to the Luttinger-Ward (LW) functional Φ15,30 and is therefore thermodynamically consistent and conserving.17,30 However, it does not respect the crossing symmetries.12 The method requires the self-consistent determination of propagators G, polarizations P = GG, self-energies Σ = −GW , and screened interactions W . The expressions for G and W are determined by the Dyson equations

(2)

where the self-energy Σ is defined with respect to a noninteracting Green’s function G0 via the Dyson equation G = G0 + G0 ΣG.

(3)

The functional formalism is useful because approximations to Φ that can be formulated as a subset of the terms of the exact Φ functional can be shown to respect the conservation laws of electron number, energy, momentum, and angular momentum by construction.16,17 In addition, Φ-derivability ensures that quantities obtained by thermodynamic or coupling constant integration from non-interacting limits are consistent.17 Functional theory therefore provides a convenient framework for constructing diagrammatic approximations in situations where a direct solution of the Hamiltonian defined in Eq. 1 is not possible. In order to discuss the formalism used in this paper, we first introduce GF2 and GW, then discuss SEET, and finally explain in detail the SEET+GW formalism.

A.

GF2

The self-consistent second order perturbation theory, GF2, is a Φ-derivable diagrammatic approximation that includes all terms up to second order in the interaction.18–26 In addition to the frequency independent Hartree-Fock terms, Σ∞ , the second order selfenergy contains X (2) Σij (τ ) = − Gkl (τ )Gmn (τ )Gpq (−τ ) klmnpq (4)
×vimqk 2vlpnj − vlpjn , where G(τ ) denotes the fully interacting Green’s function in imaginary time that is obtained self-consistently from Σ = Σ∞ + Σ(2) and the non-interacting Green’s function G0 using Eq. 3. The GF2 approach is performed iteratively, starting from a Hartree-Fock Green’s function, until Σ or the total electronic energy is converged within a predefined tolerance. In contrast to the iterative perturbation theory often used in dynamical mean field theory,27 which for the Hubbard lattice interpolates between perturbation theory and the exact solution in the limit of the separated Hubbard atoms, GF2 only relies on

B.

GW

G = G0 − G0 ΣG ,

W = v + vP W ,

(5)

where G0 and v are the bare electronic propagator and Coulomb interactions defined in Eq. 1. Self-consistent GW is only exact to first order in v, as the second-order exchange diagram is neglected. Our implementation of this approximation closely follows Refs. 12, 18, 29–31. The Green’s function is initialized using the Hartree-Fock result. We then construct the polarization P = GG and obtain W from Eq. (5). After computing the GW self-energy ΣGW = −GW , we obtain the updated G by solving Dyson’s equation, thus closing the self-consistency loop. In order to reduce the size of W , we perform a Cholesky decomposition12,32 and truncation on v. This vastly reduces the numerical effort (for related decompositions see Ref. 33). Adaptive imaginary time20 and frequency grids21 are essential to accurately represent the GW Green’s function and self-energy.

C.

SEET

The self-energy embedding theory7–11 is a conserving approximation to the Luttinger-Ward functional Φ designed to treat strongly correlated degrees of freedom. It consists of a two-step hierarchy in which the functional of the system is approximated by a solution of the entire system with a weak coupling method, which is then improved by the non-perturbative solution of correlated subsets of orbitals. In the most simple case, the strongly correlated subsets are disjoint (non-intersecting) and the SEET functional is defined as tot ΦSPLIT SEET = Φweak +

M  X i=1

 Ai i ΦA strong − Φweak .

(6)

3 Here Φtot weak denotes the approximation of the Φ functional of the entire system using a weak coupling technique. The index i enumerates the M non-intersecting, correlated subsets of orbitals Ai , and ΦAi is the functional evaluated within the orbital subset Ai , using a weak coupling or a non-perturbative method. It is much more general to consider multiple intersecting orbital subspaces. In this case, as described in Ref. 10, the SEET functional generalizes to tot ΦMIX SEET = Φweak +

M X Ai i (ΦA strong − Φweak )

(7) D.

i

−

M X

A ∩A

A ∩A

i j i j ) − Φweak (Φstrong

M X

A ∩A ∩Ak

i j (Φstrong

A ∩Aj ∩Ak

i − Φweak

SEET with GW

The general algorithmic structure of SEET is described in Refs. 7 and 9. Here, we highlight some aspects of SEET in combination with GW.

i